Consider the first 8 calculations of 7 to an exponent: 7^1 = 7 7^2 = 49 7^3 = 343 7^4 = 2,401 7^5 = 16,807 7^6 = 117,649 7^7 = 823,543 7^8 = 5,764,801 Take a look at the last digit of the first 8 calculations: 7, 9, 3, 1, 7, 9, 3, 1 The 7, 9, 3, 1 repeats through infinity. So every factor of 4, the cycle of 7, 9, 3, 1 restarts. Counting backwards from 2013, we know that 2012 is the largest number divisible by 4: 7^2013 = 7^2012 * 7^1 The cycle starts over after 2012. Which means the last digit of 7^2013 = 7
